Disorder-induced rippled phases and multicriticality in free-standing graphene

One of the most exciting phenomena observed in crystalline disordered membranes, including a suspended graphene, is rippling, i.e. a formation of static flexural deformations. Despite an active research, it still remains unclear whether the rippled phase exists in the thermodynamic limit, or it is destroyed by thermal fluctuations. We demonstrate that a sufficiently strong short-range disorder stabilizes ripples, whereas in the case of a weak disorder the thermal flexural fluctuations dominate in the thermodynamic limit. The phase diagram of the disordered suspended graphene contains two separatrices: the crumpling transition line dividing the flat and crumpled phases and the rippling transition line demarking the rippled and clean phases. At the intersection of the separatrices there is the unstable, multicritical point which splits up all four phases. Most remarkably, rippled and clean flat phases are described by a single stable fixed point which belongs to the rippling transition line. Coexistence of two flat phases in the single point is possible due to non-analiticity in corresponding renormalization group equations and reflects non-commutativity of limits of vanishing thermal and rippling fluctuations.

The study of critical elasticity of 2D crystalline membranes dates back to the seminal paper by Nelson and Peliti [1], where an idea of crumpling transition (CT), i.e. the transition between flat and crumpled phases, was put forward. A more detailed analysis of the CT and anomalous elasticity of membranes has been developed in Refs. [2][3][4][5][6][7]. The interest to the field dramatically increased after discovery of graphene [8][9][10]. A suspended graphene (for a review, see Refs. [11][12][13][14][15][16][17][18]) provides an excellent opportunity not only to experimentally verify the existing theoretical predictions for two-dimensional (2D) crystalline membranes but to challenge the theory by new unexpected experimental data. The underlying physics of CT is determined by the thermal out-of-plane fluctuations, so-called flexural phonons (FP). On the one hand, FP tend to crumple the membrane. On the other hand, the long-range interactions between FP, i.e. anharmonic effects, "iron" the membrane and stabilize the flat phase. As a result of such competition, flat and crumpled phases can exist in a clean crystalline membrane.
Along with FP, there can subsist the static, frozen deformations, the so-called ripples, caused by imperfection of the crystal lattice. Such deformations act similarly to FP and also tend to crumple the membrane as was predicted long time ago [19][20][21][22][23]. However the physics of disordered membranes with a non-trivial interplay of ripples and thermal fluctuations is much less understood as compared to the clean case. In particular, it is not even fully resolved how many phases exist in such membranes.
The competition between thermal fluctuations and ripples is of crucial importance for free-standing graphene. Indeed, the effect of the thermal fluctuations is controlled by the ratio of temperature T and the bending rigidity κ 0 . In a clean 2D membrane the CT occurs at T /κ 0 ∼ 1. In graphene, κ 0 ∼ 1 eV, so that the thermal fluctuations alone are not enough to crumple it. At the same time, recent numerical simulations of disordered graphene clearly show the CT [24]. Additional evidence for importance of disorder in graphene is provided by recent experimental measurements of anomalous Hooke's law (AHL) [25,26]. Measured scaling exponent was substantially different from the one known from numerical simulations for the clean case [27]. These experimental and numerical results imply existence of the rippled phase with properties distinct from the clean one. Previous theoretical studies of disordered 2D membranes [22,23] predict existence of the rippled flat phase at exactly zero temperature, T = 0, which is unstable with respect to the thermal fluctuations (similar conclusion has been obtained for disordered D = 4 − dimensional membrane [19][20][21]). This conclusion implies absence of the stable rippled phase, and, at first glance, contradicts to observations of Refs. [24][25][26][27]. Recently, the CT in disordered suspended graphene (DSG) was addressed in Ref. [28]. It was shown that disorder can crumple a membrane in agreement with Ref. [24]. It was also found [28] that instability of the rippled phase predicted in Refs. [22,23] develops logarithmically slow, i.e. the marginal T = 0 rippled phase controls elastic properties of DSG for T = 0 in a wide interval of length scales (see also Ref. [29]). This marginal behavior can manifest itself in experiments on AHL in graphene [25,26] as was demonstrated in Ref. [30]. However, the rippled phase should not "survive" in the thermodynamic limit even for the case of very strong disorder. Alternatively, observations of Refs. [24][25][26] can indicate the existence of a stable rippled flat phase at finite temperature. Therefore, a phase diagram of a DSG, when ripples and thermal fluctuations are competed, remains to be still established.
In this Letter, we report the phase diagram of a 2D crystalline membrane with short-ranged curvature disorder (see Fig. 1). Our main results are as follows.
• There are four distinct phases: clean/rippled flat and clean/rippled crumpled ones. There is a fully unstable, multicritical fixed point marked by M that splits up all four phases. • There is a stable fixed point F corresponding simultaneously to clean and rippled flat phases. Coexistence of two flat phases in the single fixed point reflects non-commutativity of limits of vanishing thermal and rippling fluctuations and is possible due to singularity in corresponding renormalization group (RG) equations, cf. Eq. (8). • There are two separatrices: one corresponding to the CT (red solid curve) and the other separating clean and rippled phases (blue solid curve). To obtain this results, we performed a standard 1/d c expansion [4] up to the second order, where d c is the number of FP. As was recently demonstrated [31][32][33], the second order diagrams contain ones that are not accounted by the so-called Self-Consistent Screening Approximation (SCSA) [34] which is frequently discussed as an efficient approximate scheme [29,35]. Our results represent first rigorous treatment of anharmonicity in disordered membranes within second-order in 1/d c expansion, which is not accounted for neither by SCSA, nor by other approximative schemes such as non-perturbative RG approach [36][37][38]. We demonstrate that the finite temperature instability of the rippled phase was an artefact of the first order approximation in 1/d c . Our key technical finding is that the terms of higher order in 1/d c stabilize the rippled marginal phase and lead to the appearance of the rippling transition (RT) line shown in Fig. 1.
Disorder. -There are many ways to introduce a disorder experimentally: by bombarding graphene with heavy atoms [39], by fluorination [40], or by creating macroscopical defects, e.g. artificial holes [41]. Theoretically, one classifies disorder with respect to the reflection symmetry related the two opposite sides of a membrane. An example of disorder which preserves the reflection symmetry is the so-called metric or in-plane disorder. It can arise due to the fluctuations in concentration of impurity atoms. Such short-ranged disorder is irrelevant at T = 0 in the thermodynamic limit, i.e. the clean flat phase is stable against an in-plane disorder [20,21,28] (for discussion of special case T = 0, see Ref. [42]). Therefore, we do not consider metric disorder here. Instead, we consider a random curvature disorder proposed in Refs. [19,22,23], which breaks the reflection symmetry. Such disorder naturally arises if impurity atoms are situated on one side of a membrane.
We introduce stretching factor ξ 0 , which characterizes the projective area of a membrane, ξ 2 0 L 2 , and use vectors u(x) ∈ R D and h(x) ∈ R dc to describe in-plane and out-of-plane displacements: r = ξ 0 x+u+h. Although, in the case of graphene D = 2 and the number of FP is one, we consider d c as an arbitrary parameter which allows us to develop controllable perturbation theory in 1/d c [4]. The energy of crystalline membrane consist of bending and elastic contributions [1,19,22,23] Here µ 0 and λ 0 stand for the Lamé coefficients. The last two terms in the right hand side (r.h.s.) of Eq.
(1) describe in-plane elastic energy with the strain tensor u αβ = (∂ α u β + ∂ β u α + ∂ α h∂ β h)/2. Quenched random curvature is added via a zero-mean Gaussian random vector β [19,22,23]. Strength of disorder is con- Anomalous elasticity. -Although the CT cannot be observed in a clean graphene, the thermal fluctuations around the flat phase lead to highly non-trivial anomalous elastic properties in membranes of size above the so-called Ginzburg length denotes the bare (ultraviolet) value of the 2D Young's modulus. Due to an anharmonic coupling between the in-plane and out-of plane elastic modes the bending rigidity increases in a power law manner for L L * with a certain critical exponent η: κ ∝ κ 0 (L/L * ) η . For graphene L * ∼ 1 ÷ 10 nm, so that realistic flakes of graphene are in the regime of anomalous elasticity and show a number of highly non-trivial phenomena already verified experimentally, such as mentioned above AHL [5, 6, 25-27, 30, 43, 44], negative thermal expansion coefficient [45][46][47][48][49][50][51], power-law scaling of the phonon-limited conductivity [52][53][54], etc.
Crumpling transition. -The scaling equation describing dependence of the stretching factor on a membrane size L reads [28] where T = T /κξ 2 and B = b/ξ 2 are rescaled amplitudes of the thermal and disorder-induced fluctuations. The CT occurs when ξ turns into zero at a finite length scale, while in the flat phase ξ(L → ∞) > 0. Both the thermal and rippling fluctuations tend to crumple the membrane.
In the clean case, B = 0, the power law dependence of T on L results in the crumpling transition at a very high temperature T c = 4πηκ 0 /d c , which is unreachable for graphene.
In a disordered membrane scaling of T and B is more intricate than a power law. There is a CT curve in the plane (T , B) which was found in Ref. [28] by using first order expansion over 1/d c . Next, we demonstrate that higher order terms in 1/d c lead to appearance of the rippling transition line (blue line in Fig. 1).
Replicating fields u and h, integrating over u and performing averaging of the replicated partition function over disorder, we obtain the effective free energy [28].
whereκ ab = κ 0 δ ab − f 0 J ab with matrixĴ having all entries equal to one, indices a, b = 1, . . . , N enumerate replicas, and f 0 = b 0 κ 0 /T . Anharmonicity of FP result in a renormalization of the parameters of F dis . The necessary information can be extracted from the exact two-point Green's function h j (−k) ≡Ĝ ab (k)δ ij where the average is with respect to the free energy (3). The quadratic part of F dis determines the bare Green's functionĜ ab (k) = T (δ ab + f 0 J ab )/(κ 0 k 4 ). At first, the screening of the interaction between flexural phonons should be taken into account via RPA-type resummation (see Fig. 2). The screened interaction becomes independent of Y 0 for q < d c (1 + 2f 0 )/L * [28] and behaves as q 2 /d c as q → 0. Using this screened interaction we can construct the regular perturbation theory in 1/d c for the self-energyΣ (see diagrams in Figs RG flow.
-The corresponding RG equations can be written in the following form.
We emphasize that RG equation for f decouples, while κ and b are slave variables. The RG functions can be expanded as β (2) = − 73 + 803f + 3667f 2 + 8517f 3 + 9278f 4 +3420f 5 + 186f 6 − 68ζ(3) 1 + 11f + 49f 2 +111f 3 + 128f 4 + 58f 5 + 6f 6 27(1 + 2f ) 6 , The functions β (2) and η (2) κ have finite limit at f → ∞. The RG flow for f has three fixed points: 0, ∞, and f * ≈ 8.5 d c (see the inset to Fig. 1) [57]. At the fixed point f = 0 which is stable in the infrared the bending rigidity and disorder variance acquires the power law scaling, κ ∼ L η , b ∼ L −η where [33] The other infrared stable fixed point is located at f = ∞. We note that within the first order in 1/d c this fixed point is marginally unstable [22]. At f = ∞ the bending rigidity and the disorder variance has also power-law scaling with momentum, The fixed point at f = f * is unstable in the infrared and is characterized by the exponent of divergent correlation . From Eqs. (2) and (4) we find the RG equations governing the flow of parameters T and B.
The corresponding flow diagram is shown in Fig. 1 and B * = f * T * (see Fig. 1). This multifractal fixed point has two unstable directions: along the CT curve, which demarks flat and crumpled phases, and along the RT line, B = f * T , which splits up clean and rippled phases and connects M and F . The scaling along these two separatrices are controlled by the critical exponents ν and 1/η κ (f * ), respectively. We emphasise the striking resemblance of our RG flow diagram with the one for the random bond Ising model [59]. The RT line corresponds to the so-called Nishimori line [60]. Discussion and conclusion. -Our key result is the demonstration of the ripples stabilization by sufficiently strong disorder. More precisely, two transitions occur with increasing the disorder at fixed other parameters. For T < T * (see Fig. 1), the first transition corresponds to the stabilization of ripples, while the second one is the CT. On the contrary, for T > T * , the CT happens before the stabilization of ripples. Our phase diagram suggests also a possibility of the RT with decreasing temperature at the fixed disorder. As a very interesting subject for further research, we expect that the phase diagram is even reacher in the case of long-ranged disorder [61].
In the course of derivation of RG Eqs. (8) we neglected the term ∂ α u∂ β u in the expression for the strain tensor u αβ . It can be shown [55] that this approximation is justified for B, T 1. The terms ∂ α u∂ β u provide additional contributions to Eqs. The relevance of our theory for realistic graphene membranes is supported by the numerical simulations [24] where the CT with increase of disorder was clearly seen and the fractal dimension of the crumpled membrane was reported. A detailed comparison of our theory with Ref. [24] is however not possible due to a lack of simulations at various temperatures.
To conclude, we predict existence of two disorderdominated rippled phase (flat and crumpled) in a disordered crystalline membrane with a short-range disorder. By using fully controlled standard 1/d c expansion, we derive coupled RG equations for the bending rigidity and disorder strength and establish the phase diagram of a generic crystalline membrane (see Fig. 1). We demonstrate existence of the multicritical point (M), the singular stable point (F), where the rippled flat and clean flat phases coexist, and the rippling transition line connecting these two fixed points.
We thank I. Gornyi

Disorder-induced rippled phases and multicriticality in a free-standing graphene
In this Supplementary Material we present derivation of the RG equations (4) of the main text.

SELF-ENERGY CORRECTION
The interaction between flexural phonons modifies the Green's function. The exact Green's function can be written as follows (in the replica limit N → 0):Ĝ As well-known, before constructing the perturbation theory in the interaction between flexural phonons it is important to take into account screening of this interaction by the flexural phonon themselves. This screening (see Fig.  2b of the main text) is determined by the bare polarization operator Here we introduced for a brevity the following shorthand notation: k ≡ d 2 k/(2π) 2 . Summation of the geometric series shown in Fig. 2b of the main text yields the screened interaction We mention that the screened interaction at small momenta, q q * = 1 + 2f 0 /L * becomes independent of the Young modulus Y 0 and proportional to 1/d c .
Contribution of the first order in 1/dc The self-energy correction of the first order in 1/d c is given by the diagram in Fig. 2a of the main text. It can be written as:Σ where For m = 0, 1 they are given explicitly as follows In the limit k/q * 1 and N → 0, we find where q * =q * exp[1/4 + γ 1 /(2α 1 )] and q * =q * exp(−1/4) and Contribution of the second order in 1/dc In this subsection we present results for the contribution of the second order in 1/d c to the self-energy (see diagrams in Fig. 3

of the main text)
Diagram Fig. 3a The corresponding contribution to the self-energy has the following form Σ (2,a) Computing the integrals over momentum q in the same way as in Ref.
Integration over momenta can be performed in the same way as in Ref.